guest lecture, october 3 2016 - neil sloaneneilsloane.com/doc/math640fall2016.pdf · •f is a...
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Math 640: EXPERIMENTAL MATHEMATICS
Fall 2016 (Rutgers University)
Guest Lecture, October 3 2016
Neil J. A. SloaneMathematics Department, Rutgers
andThe OEIS Foundation
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Outline• 1. Quick overview of OEIS
• 2. Some recent sequences of great interest
• 3. Combinatorial games: Wythoff’s Nim
• 4. Beatty sequences; Class project 1
• 5. Morphisms: Class project 2
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Section 2.
Some recent sequences of great interest
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OEIS.orgThe new poster,
on the OEIS Foundation web
site
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Peaceable QueensA250000
Peaceable coexisting armies of queens: the maximum number m such that m white queens and m black queens can coexist on an n X nchessboard without attacking each other.
0, 0, 1, 2, 4, 5, 7, 9, 12, 14, 17, 21, 24
4 X 4 11 X 11Monday, October 3, 16
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A250000
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(Michael De Vlieger)Monday, October 3, 16
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x = 1/4, y = 1/3, density = .146, Optimal?
Peter Karpov
A250000
(Lower bound .141)
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How many ways to draw n circles in (affine)
plane?
A250001
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No. of arrangements of n circles in the plane
1, 3, 14, 173, 16942
Jonathan Wild
What if allow tangencies?
A250001
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a(4) = 173
https://oeis.org/A250001/a250001_4.pdf
Amazing pictures!
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Bingo-4A273916
Demonstrate by looking at to OEIS entryLovely new question from China
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Three other new seqs.
• A274647: Variation on Recaman, A5132. Again, does very number appear? We don’t know, but maybe this version is easier. Look at graph
• A276457: Very nice problem. Graph, play!!
• A276633: Nice problem. Look at graph, listen.
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Section 3
Combinatorial games: Wythoff’s Nim
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Combinatorial GamesWhoever takes the last coin wins.
Nim: You win if you leave your opponent 3 piles of 1,2,3, or 2 piles 1,1 (say)
These are winning positions, but to avoid ambiguity they are called P-positions, meaning you, the PREVIOUS player, wins. Any other position is an N-position, meaning NEXT
player wins.
In Nim, a set of piles of sizes a,b,c,... is a P-position iff mod 2 sum a+b+c+... (no carries) is 0
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Wythoff’s Nim (1907)Two piles, of sizes a and b
- can remove any number (>0) from one pile- can remove equal numbers (>0) from both piles
P-positions are:
(0,0), (1,2), (2,1), (3,5), (5,3), (4,7), (7,4), (6,10), ...What are these numbers? (a_n, b_n), a_n <= b_n.
a_n: 0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21,0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34,b_n:
A201A1950
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Theorem 1 (Wythoff)
a_n = mex { a_i, b_i, i<n }b_n = a_n + n
i. Cannot move from (a_n, b_n) to (a_i, b_i), i<n
Proof:
ii. CAN move from (a,b) not of form (a_n, b_n) to a position of the form (a_n, b_n).
Must show:
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Section 4
Beatty sequences; Class project 1
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Beatty SequencesDefn.: If x > 0 is real, the sequence
S_x := { floor(nx): n=1,2,3...}is called the spectrum of x
Theorem 2 (Beatty1927)
If a, b > 1 are irrational and 1/a + 1/b = 1then S_a and S_b are disjoint and include
every positive integer.
Proof will be given.
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Theorem 3 (Wythoff)
Winning positions in Wythoff’s Nim are (a_n, b_n) and (b_n, a_n), where
a_n = floor(n t), b_n = floor(n t^2)
where t = (1+sqrt(5))/2 is the Golden ratio.
Sketch of proof
t^2 = t+1, so 1 = 1/t + 1/t^2, so S_t and S_t^2are complementary sequences (Th. 2).
Claim: [nt] = mex { [it], [it^2], i<n } (induction)and [n t^2] = [nt] + n.
Result now follows from Th. 1
A201 andA1950
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Ron Graham’s Test for a Beatty Sequence
Sloane and Plouffe, Encyclopedia of Integer Sequences, Academic Press 1995; Chap 2, How to handle a strange sequence
Graham and Lin, Spectra of numbers, Math. Mag., 51 (1978)
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Class Project 1Implement Graham’s test for Beatty sequences in Maple,
run it on all plausible sequences in OEIStry to discover Beatty sequences that are not
at present identified as Beatty.
[Can rule out non-increasing or neg. terms, keyword sign, tabl, tabf, frac, cons, word, dumb, etc.]
Note: Compressed versions of data lines and of name lines are on OEIS Wiki, see section on
“Compressed Versions”
If find a plausible candidate for a hitherto unknown Beatty sequence, check Graham’s test on the b-file, if there is one. If still passes test, try to find proof.
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• data lines: stripped.gz, 278347 lines, 12 Meg
• name lines: names.gz
Compressed versions of the OEIS
(see https://oeis.org/wiki/Main_Page ,section “JSON Format and Compressed files”)
If you find that a sequence is (or even seems to be)a Beatty seq., add a (signed) comment to the entry!
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Section 5
Morphisms: Class project 2
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Combinatorics on WordsCanonical example: the Fibonacci word W
Define “Morphism”: 0 01, 1 0
The Fibonacci word W is the trajectory of 0 under this map
W_1 = 0, W_2 = 01, W_3 = 010, W_4 = 01001
W_5 = 01001010, ..., W_oo = W
W_{n+1} = W_n W_{n-1} |W_n| = Fib_n
A3849
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Back to lower Wythoff sequence a_n = floor(nt):
1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, ...2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, ... A14675
A201
which is a version of the Fibonacci word !
1 2, 2 21Morphism is
Upper Wythoff sequence b_n = floor(nt^2):
First differences are3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3 A76662
which is the same but over alphabet {2,3}
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Morphic SequencesSpecial case, there is a more general definition,
see Allouche and Shallitt, “Automatic Sequences,”Cambridge Univ. Press, 2003
S is morphic if it is a sequence over a finite alphabetC = {c_1, c_2, ..., c_r}
which is the trajectory of c_1 under some morphism f
f(c_1) = ...f(c_2) = ...
...f(c_r) = ...
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which means that S is an infinite sequence (or word) such that f(S) = S
Allouche and Shallitt, Theorem 7.3.1, p. 216give n.a.s.c. for f, S to satisfy f(S) = S
Class Project 2Write a Maple program to test a given sequence
(over a finite alphabet) is morphic,and search the OEIS for sequences that might be
morphic, but are not yet identified as such.
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• Use compressed version of OEIS database
• Can eliminate many right away
• There is an entry in the OEIS Index for “fixed points of mappings”
• If find a candidate, test the b-file
• If stll looks morphic, try to prove it
• Either way, add comment to the entry: This [is / appears to be] the trajectory of x under the morphism f = ...
Notes:
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• f is a uniform morphism if all f(c_i) have same length [Thue-Morse A10060 is uniform, 0 to 01, 1 to 10]
• most interesting morphisms not uniform
• the Index entry mentioned above gives many interesting example
• the Index entry is old and needs to be brought up to date. Please help!
• Add following link to all morphic sequence:
• <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
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